Ecmandu wrote:I was a step ahead of you... 2 - (-2) is a double subtraction, which inverts what I said about subtraction itself. I'll just forget that you ev n bothered to say subtraction is addition.

Opposite direction to the opposite direction is the same direction.... so?

Like I said, don't look to me for confirmation that subtraction is addition, just learn how computers work. Subtraction is addition... but in the opposite direction (same magnitude). Does that mean subtraction = addition?

Seriously, how are you misunderstanding me, maths and computers so badly? One step ahead? If ahead means in the opposite direction, sure.

Of course to subtract, you must be adding something....(duh) the subtraction, but this does not make the subtraction an added property in the sense that 2-2 = 4. 2-4 has a span of 4, rendering to 2 a zero, which can only happen with opposites, not with 2-4... can you understand that?

Ecmandu wrote:this does not make the subtraction an added property in the sense that 2-2 = 4.

What? You think I'm saying subtraction is addition? That's the only way 2-2=4, and that's not what I mean. Be clear what you mean.

Ecmandu wrote:2-4 has a span of 4

Span? What? You mean from 2 to -2 the magnitude moved is 4 because you -4? Obviously....

Ecmandu wrote:rendering to 2 a zero, which can only happen with opposites, not with 2-4... can you understand that?

Can you explain properly, please? Then maybe you would make sense.

surreptitious75 wrote:I/0 is supposed to be infinity but that cannot be true because 0 times infinity is still only 0

This raises the interesting mathematical question of what if anything is greater than infinity

Assume for example that I/0 = X then what value would X have ? It would have to be infinity plus

I addressed this in a previous post if you've not seen it yet.

Infinity plus isn't really possible, but mathematicians like to see what happens if you denote an impossible concept and see if using it as a possible concept results in anything interesting or useful - such as complex numbers (e.g. "i"). But really, since you never get to infinity, you're equally not going to get to greater than even that, so they're the same concept. In case you're not aware, infinity isn't actually a number, it's beyond numbers. You can't really treat it like one, but like I said about mathematicians, you're free to see what happens if you try. Just be prepared for results that make no sense and end in contradictions.

Like I also said, 1/0 isn't infinity or infinity plus, it's undefinable for the reasons I already explained. 1/0 is just an invalid expression: it doesn't matter how infinite you make infinities, this remains true.

Subtract means to take away from so logically that can not be classed as addition2 - 2 means 2 taken away from 2 not 2 added on to 2 before it is then taken awayA negative integer subtracted from a positive integer always leaves a smaller one

surreptitious75 wrote:Subtract means to take away from so logically that can not be classed as addition2 - 2 means 2 taken away from 2 not 2 added on to 2 before it is then taken awayA negative integer subtracted from a positive integer always leaves a smaller one

Nobody is saying 2 - 2 means adding 2 to 2 before taking it away. You just "add backwards" only: that's the same as "taking away" - and you don't do one and then the other!

It's a matter of perspective. For example, In your computer a negative number is simulated by "wrapping" around at the mid point (known as the two's complement). It goes from 0 to just before half the maximum number you can make from the number of bits you're using, then it goes to negative half maximum number and ascends back to -1. But the bits themselves just look like they're increasing the whole time. This cyclical interpretation of addition is how subtraction is derived in your computer - everything in your computer is derived from addition.

You might even think of addition like the magnitude moved, and subtraction as an afterthought on the direction you moved in respect to the number you started from. All of maths is addition with or without a twist on it. Note this is not saying that subtraction is addition, subtraction is addition in the other direction. If you swap which direction you're thinking of as "increase", then you'll realise subtraction is just the opposite of whichever that direction happens to be - so long as you're keeping everything consistent and swapping the direction of everything when you do so.

surreptitious75 wrote:Ecmandu : 2 - 2 is both subtraction and opposite : subtraction because 2 is being taken away from 2 and opposite because - 2 is the opposite of 2

Subtraction, yes. Opposite? In direction, yes. But the magnitude is the same in each case (2). NOT is opposite in all respects, negative is only opposite in direction. Integers are scalars, with only magnitude, but signed integers (e.g. negatives) are vectors, which need both magnitude and direction. You're thinking selectively, cherry-picking, to try and force this "NOT = negative" nonsense to appear to work. More likely though, you're thinking ignorantly because you simply don't have the education on the matter to realise where you're going wrong.

Ecmandu wrote:Ahh yes the 1/ zero times = 1Zero divided 1 time, is indivisible, so it just equals zero and not infinity or undefined.

What? Who said 1/0 = 1?

Nobody.

0/1? How many times do you need to subtract the denominator (1) from the numerator (0) until you get to 0? Well 1 is too much, because that gets you past your goal to -1. Half? Still the same problem. Only zero times gets you to 0. That's why 0/1 = 0. 0/1 is defined, it is indeed not infinity nor undefined.1/0? How many times to you need to subtract the denominator (0) from the numerator (1) until you get to 0? No number of times will ever get you lower than 1, not even an infinite number of times. Infinitely subtracting 0 from 1 still keeps you at 1. It's an invalid expression, it's undefinable, it's not infinite, it's not anything, not even nothing.

Ecmandu wrote:It's amazing how the elite have trained people to parrot stuff that cannot possible be true !

It's amazing how the untrained or dumb (still not sure which of the two you are) make up stuff that cannot possibly be true without even realising, even when shown incontrovertibly what the truth actually is over and over.

Ecmandu wrote:Like I said before, you didn't use a negative to make your proof in the case of 2, you used a double negative. 2-(-2). So it's not a proof of negative and shows that negative itself means opposite ..,

You never did clarify whatever the hell you were on about with the double negative thing... how can I possibly comment on an imagined interpretation of what I didn't say about a pile of nonsense that you came up with? I've demonstrated in several ways by now how negative doesn't mean opposite. Opposite direction for a vector quantity (e.g. -2 compared to +2) doesn't mean the complete opposite because the magnitude is the same. How are two things with the same magnitude the opposite? For the opposite, you need the NOT operator (jesus, how many times do I have to explain this?)!

Ecmandu wrote:Ahh yes the 1/ zero times = 1Zero divided 1 time, is indivisible, so it just equals zero and not infinity or undefined.

What? Who said 1/0 = 1?

Nobody.

0/1? How many times do you need to subtract the denominator (1) from the numerator (0) until you get to 0? Well 1 is too much, because that gets you past your goal to -1. Half? Still the same problem. Only zero times gets you to 0. That's why 0/1 = 0. 0/1 is defined, it is indeed not infinity nor undefined.1/0? How many times to you need to subtract the denominator (0) from the numerator (1) until you get to 0? No number of times will ever get you lower than 1, not even an infinite number of times. Infinitely subtracting 0 from 1 still keeps you at 1. It's an invalid expression, it's undefinable, it's not infinite, it's not anything, not even nothing.

Ecmandu wrote:It's amazing how the elite have trained people to parrot stuff that cannot possible be true !

It's amazing how the untrained or dumb (still not sure which of the two you are) make up stuff that cannot possibly be true without even realising, even when shown incontrovertibly what the truth actually is over and over.

Ecmandu wrote:Like I said before, you didn't use a negative to make your proof in the case of 2, you used a double negative. 2-(-2). So it's not a proof of negative and shows that negative itself means opposite ..,

You never did clarify whatever the hell you were on about with the double negative thing... how can I possibly comment on an imagined interpretation of what I didn't say about a pile of nonsense that you came up with? I've demonstrated in several ways by now how negative doesn't mean opposite. Opposite direction for a vector quantity (e.g. -2 compared to +2) doesn't mean the complete opposite because the magnitude is the same. How are two things with the same magnitude the opposite? For the opposite, you need the NOT operator (jesus, how many times do I have to explain this?)!

Addition is a different operator than multiplication, multiplication is not addition squared. In the same sense division is not the same operator as subtraction, it is not subtraction squared.

The reason I point this out is because division is reciprocal to multiplication. If 1 is divided 0 TIMES!! It's still one because it was never divided. If zero is divided 1 TIME, it's still zero, because it's not there to be divided.

I'll be blunt with you.

We are speaking about different modalities of mathematics... yours is not linguistically salvageable, mine is.

Ecmandu wrote:Addition is a different operator than multiplication, multiplication is not addition squared. In the same sense division is not the same operator as subtraction, it is not subtraction squared.

The reason I point this out is because division is reciprocal to multiplication. If 1 is divided 0 TIMES!! It's still one because it was never divided. If zero is divided 1 TIME, it's still zero, because it's not there to be divided.

I'll be blunt with you.

We are speaking about different modalities of mathematics... yours is not linguistically salvageable, mine is.

Yet you accuse me of nonsense.

Addition squared? Subtraction squared?? What the hell are you on about now?!

What is the point of me explaining things if you either never read them/ignore them/forget them??

I already explained how division is derived from subtraction and therefore by addition, and in the same way multiplication is derived from addition. And no, you don't need to "square" or "root" anything before you want to put any more nonsense in my mouth.Multiplication is the amount of times you need to add something to itself to get to the answer e.g. 3x4 is 3 added to itself 4 times - that's why it's called 3 times 4.Division is the amount of times you need to subtract the denominator from the numerator to get to zero (or add the denominator to itself to get to the numerator) e.g. 12/3 = subtract 3 from 12 to get to zero = 4 times or add 3 to itself to get to 12 = 4 times.They're all grounded in addition, and interestingly enough since you mention them - but probably beyond you, squares and roots also derive from addition too. This is literally how your computer works - the very thing you're using to argue against the concepts that make it work... I'm not making this up, it's very real and in your life everywhere. I'm using a different "modality"??? No, I'm just telling you what maths actually is and where it actually came from. You're making up nonsense and putting nonsense into the mouth of others to try and make something ignorant and clearly flawed seem relevant.

I mean, here you are trying to say that 1/0 is 1. Yes you can use language to misleadingly represent something like division so it sounds like it doesn't mean what it means. The word for this is Sophistry.

I rest my case, there's no way you can salvage what you're trying now with any amount of linguistics. Good bye.

Silhouette wrote:Multiplication is the amount of times you need to add something to itself to get to the answer e.g. 3x4 is 3 added to itself 4 times - that's why it's called 3 times 4.

So how do you multiply pi times the square root of 2 by your definition?

Bonus question: In the complex numbers, how do you show that i^2 = -1? How many i's is that? How do you reduce this problem to addition?

Fortunately for computers the solution to multiplying irrational numbers is forcibly simplified by the number of bits you have to work with. Thereby, you apply the same reduction to addition that I explained to an approximation of said irrational numbers i.e. to a limited number of decimal (or binary in this case) places - as though they were rational numbers. To deal with (non-integer) numbers that have decimal points, variables of "float" or "double" type are used to represent numbers in an alternative way to allow more decimal places to be dealt with, even though the same number of binary places are being used - at the cost of accuracy. "Float" is short for floating decimal point, which hints at how the alternative representation manipulates bits to this end, although that's not the only trick used. Still though, the limited number of bits nevertheless results in irrational numbers being dealt with as though they were rational - and the problem posed to mathematicians of irrational numbers is circumvented artificially.

Even outside of the world of computers, the same constraints are forced by practicality. Most vulgarly, for engineers and others who use irrational numbers in calculations that apply to everyday scales, the use of more decimal places quickly becomes redundant, meaning they too use irrational numbers like they were rational. But even for quantum physicists performing experiments at the quantum level, the use of more decimal places also becomes eventually redundant beyond a certain only slightly further threshold and the same truncation is resorted to - and everything still boils down to a twist on addition.

But what about for theoretical physicists and pure mathematicians? Their solution is even simpler: algebra. Got an irrational number? Use a letter to denote it (in exactly the same way that numbers denote integer quantities). This preserves the implied infinity of decimal places throughout all calculations, sometimes being cancelled out, sometimes being eliminated by certain properties of irrational numbers when used in, for example geometry. Consider that famous identity by Euler: "e^(pi)i = -1". This equation is most simply shown on an 2D graph of a (complex) unit circle that plots its imaginary component against its real component, where pi is in radians and e is the base of natural logs as standard. Here we have 3 non-real numbers that when related to one another in a specific way amount to a real number, an integer no less.

Technically, "i" is just another kind of anomaly like irrational numbers that presents itself when building up maths from addition upwards - and to deal with that, in answer to your bonus question, the same solution of algebra is used. Mathematicians have to bring everything back into the realm of "boils down to addition" in order to deal with it - even the apparent exceptions that emerge from building maths in such a way. The whole point of maths is that it boils down to the simplest possible concepts, such that it is robust and consistent throughout.

In short: the practical solution is to truncate to a rational approximation, the theoretical solution is to use algebra.

Thank you for asking the first not-profoundly-stupid question in this thread for far too long.

Silhouette wrote:Thank you for asking the first not-profoundly-stupid question in this thread for far too long.

I appreciate the kind words. So now if I were to say "You're all wet," or, "You have no idea what you're talking about," I'd feel guilty. So I'll just point out a few things. I could write a lengthy post but I'll keep this mercifully short and just list some bullet iitems.

* The mathematical real numbers \(\mathbb R\) are not anything remotely like the floating point numbers as described by IEEE-754. Yes I know what that is and they are not the real numbers. You seem to have studied computers and not math. Everything you say about computer arithmetic may be true, but totally irrelevant. The question was not, "Can computer multiplication be reduced to addition?" That's completely different question to whether real number multiplication can be so reduced.

* Likewise your off-topic remarks about engineering math, and all practical operations involving real numbers being reducible to rationals. Of course that's true, but equally irrelevant. We're talking about the mathematical real numbers. You know, the ones that require infinitely much information to represent.

* To take you entirely out of the realm of computers, how do I multiply two noncomputable real numbers? Those are the numbers that can not be approximated by a program or a Turing machine. Most reals are noncomputable, as you can see from noting that there are uncountably many reals but only countably many Turing machines. How do you reduce the multiplication of two noncomputable reals to "repeated addition?" The idea is absurd on its face. You could not reduce a noncomputable real to a rational approximation with any finite amount of computing power or memory no matter how large, if your approximation is required to be computable.

* Your handwaving about "algebra" is nonsense. Yes I'm mindful that you said something nice to me. I regret not being able to respond in kind other than to note that you know a lot about computers and engineering math but sadly nothing about math. Your remarks in this area were vague. What do you mean that "algebra" shows that multiplication is reducible to repeated addition? On the contrary. In algebra, multiplication is a map from pairs of real numbers to real numbers, satisfying the usual field axioms. There's nothing in the field axioms about multiplication being repeated addition.

* Likewise your handwaving that " "i" is just another kind of anomaly like irrational numbers ..." Please friend, I really appreciate that you complimented me. I'm going to hate myself in the morning for being so churlish as to say this, but you just embarrassed yourself.

It's ok that you studied computers and engineering instead of math. But be humble about what you don't know.

If you'd like me to expand on anything I said, please ask. Like I said I could have written a lot more.

One more point. Here is what you meant to say, if you'd studied math: "Well we can define multiplication as repeated addition in Peano arithmetic; then we can lift multiplication to rationals using the standard field of quotients construction; and then we can lift multiplication to the reals by taking limits of Dedekind cuts of rationals; and in this way, although real number multiplication is NOT in any meaningful sense repeated addition, we can indeed find multiplication defined as repeated addition a long way back in the chain of the construction of the real numbers." If you said that, it would be the right answer. I'll leave the definition of multiplication of complex numbers to you.

Another simple way out would have been to say that we can view multiplication as repeated addition for natural numbers but not for rationals, reals, or complex numbers. That would be sensible. But your changing the subject to computers was wrong, because floats are not real numbers. Maybe they just don't tell the CS students that. But you should realize, there's a LOT they don't tell the CS students about math. They run you through "Discrete math" and consider your mathematical education done. It's a crime.

Serendipper wrote:I never heard of that, but I've seen -0 mean "to the left of zero" and +0 meaning "to the right of zero" which is the smallest infinitesimal representationally possible.

There could be no such thing. In the standard real numbers there are no infinitesimals and there is no smallest positive real. The proof is that if you tell me x is the smallest positive real, then x/2 is a positive real that's smaller.

But suppose you say, well I've heard of the hyperreals and nonstandard analysis, in which you can do calculus but there are genuine infinitesimals. What about that? It's true that there are systems with infinitesimals. But there is no smallest positive infinitesimal, and the proof I already gave for the real number case works in the nonstandard case too. That's because you can always divide an infinitesimal in half.

So there is no smallest positive infinitesimal even in number systems that have infinitesimals.

I would argue from the perspective of reality (rather than abstract mathematics where you are perfectly correct) which implies physical limitations on the representation of numbers where those lying without are referred to as "dark numbers", which are those that cannot be represented within the observable universe even if written at the planck scale. There are certainly numbers that cannot be said to exist in any meaningful way other than trivial abstraction.

Math is merely a construct and not necessarily representative of what we call reality nor constrained by any limitations thereof, so our definition of the symbolic nomenclature must be in the context of what "just to the right of zero" means to the one using the information. In abstractness, there is no number just to the right of zero, but in counting apples, there is: 1. On a tape measure, the gradations are generally limited as no carpenter implementing a role-up tape could make any meaningful use of 1/100th of an inch when framing a house.

Pragmatically, "to the right of zero" would mean the smallest number you find meaningful and its symbol would be 0+.

Ecmandu wrote:Zero is used as both referring to the lack of numbers, but it is also the base placeholder.

Negative zero is every number including zero

Technically that would be the "NOT" operator rather than the negative operator.

Yes I think you're right.

+1 is the opposite of -1, but not-1 means any number that is not 1.

Xunzian wrote:1/+0= +infinity

1/-0= -infinity

It's been a while and I was never that great at calc but for f(x)=1/x as x->-0, f(x)-> -infinity and as x->+0, f(x)-> +infinity

That describes the graph of f(x)=1/x pretty well to me.

1/0 isn't infinity, it's undefinable.

Infinity is undefinable

I'm going to reply to this by simply stating: the issue at hand is zero, and strange properties of zero relative to non zero numbers.

To use non zero numbers as analogies is to miss the argument completely.

Actually that charge silloutte is is brining against me is not "sophistry" as claimed, but rather "equivocation".

I stated outright that these are modalities of math and that mine is grammatically correct ... to silhouette, this is putting words on people's mouths, to which I replied, these are different modalities.

Serendipper wrote:I would argue from the perspective of reality (rather than abstract mathematics where you are perfectly correct) which implies physical limitations on the representation of numbers where those lying without are referred to as "dark numbers", which are those that cannot be represented within the observable universe even if written at the planck scale. There are certainly numbers that cannot be said to exist in any meaningful way other than trivial abstraction.

I perfectly well stipulate that I know nothing of reality. I'm not even convinced there is one. We could be a brain in a vat, an experiment in an alien grad student's AI lab, or a program running in God's own Turing machine. I try to clarify mathematical issues when they come up. But I certainly have no idea of what's "out there." Personally I don't think the standard real numbers are literally how reality works. To me, the real numbers are a wonderfully interesting mathematical abstraction that probably have no relation at all to the world. Or maybe they do. It's fun to talk about.

Serendipper wrote:Math is merely a construct and not necessarily representative of what we call reality nor constrained by any limitations thereof,

Absolutely. I feel like I spend my life trying to explain this to people online. Then other people try to explain it to me, as if they think I'm arguing the opposite.

Serendipper wrote: so our definition of the symbolic nomenclature must be in the context of what "just to the right of zero" means to the one using the information.

Ah sorry part with you here. When we try to formalize our vague intuitive notions of infinitesimals, we find that there is not a smallest positive real number in ANY conceivable model of the real numbers, in particular the hyperreals, which are everyone's go-to example of a system of numbers with infinitesimals.

So if you are making a philosophical point, you still can't violate known math. If I'm understanding you correctly.

Serendipper wrote: In abstractness, there is no number just to the right of zero, but in counting apples, there is: 1. On a tape measure, the gradations are generally limited as no carpenter implementing a role-up tape could make any meaningful use of 1/100th of an inch when framing a house.

Ok.

Serendipper wrote:Pragmatically, "to the right of zero" would mean the smallest number you find meaningful and its symbol would be 0+.

Ah I see where you're going with this. No, there is no smallest number or infinitesimal just to the right of zero. Because if you say there is, and you call it x, then I'll just point out that 0 < x/2 < x and x was not the smallest after all.

Ecmandu wrote:I'm going to reply to this by simply stating: the issue at hand is zero, and strange properties of zero relative to non zero numbers.

But zero is nothing... absence of things. How can nothing have properties?

Actually that charge silloutte is is brining against me is not "sophistry" as claimed, but rather "equivocation".

Sophistry implies mal-intent, right?

I stated outright that these are modalities of math and that mine is grammatically correct ... to silhouette, this is putting words on people's mouths, to which I replied, these are different modalities.

Serendipper wrote:Pragmatically, "to the right of zero" would mean the smallest number you find meaningful and its symbol would be 0+.

Ah I see where you're going with this. No, there is no smallest number or infinitesimal just to the right of zero. Because if you say there is, and you call it x, then I'll just point out that 0 < x/2 < x and x was not the smallest after all.

That's math, brotha.

You're perfectly correct in math, but in what we colloquially regard as reality there are limits to numbers.

Serendipper wrote:You're perfectly correct in math, but in what we colloquially regard as reality there are limits to numbers.

Check out this video.

Some video some guy made is not proof of anything. Even if I spent 30 minutes of my life watching it, if I disagreed with anything he said I'd just be arguing with some guy who made a video. What is the point of that?

If you can state an argument relevant to my post we can have a discussion. You and I, not me and some guy on the Internet with an opinion. I can't argue with every Youtube video that's out there. I'd have to make my way through the flat earthers and the moon landing deniers long before I got to the infinity opinionators.

I am a professor of mathematics at UNSW Sydney. I was educated at Adam Scott High School in Peterboro Ontario, Richmond Hill High School in Richmond Hill Ontario, University of Toronto (BSC 1979) and Yale University (PhD 1984). I taught at Stanford University (1984-1986) and the University of Toronto (1986-1989) before coming to UNSW (University of New South Wales), Sydney, in 1990.http://web.maths.unsw.edu.au/~norman/

My PhD thesis was Quantization and Harmonic Analysis on Nilpotent Lie Groups. I have worked in representation theory, harmonic analysis, combinatorics, and geometry. I have developed finite hypergroups and duality, Pell's equation and Diophantine equations, and introduced Rational Trigonometry and chromogeometry (download my book here!) I have reformulated hyperbolic geometry to make it more algebraic, general and beautiful. I have a YouTube channel: njwildberger with 600+ math videos.https://www.researchgate.net/profile/Norman_Wildberger

It would be like calling Stephen Hawking "some guy".

Even if I spent 30 minutes of my life watching it, if I disagreed with anything he said I'd just be arguing with some guy who made a video. What is the point of that?

I told you FWD to 10:00. It's like 1 min and if you're interested, then listen to the other. If not, then don't.

Stay ignorant. I don't care.

It's as if you said "If you do not feed me the information I want in the form I want it, I'm going to stay uninformed! So there!" <-- Not scary.

If you can state an argument relevant to my post we can have a discussion.

I did. The universe presents a limitation on math. Watch the accomplished and passionate mathematician say it again on the video along with his reasoning.

You and I, not me and some guy on the Internet with an opinion. I can't argue with every Youtube video that's out there. I'd have to make my way through the flat earthers and the moon landing deniers long before I got to the infinity opinionators.

You went so far into left field that you're in the parking lot.

Why would you have to watch flat-earthers and moon landing deniers before watching 1 min of the video I posted?